Consider a Turing machine which reads its input and then immediately stops. This Turing machine always halts on every input, but the running time is unbounded: the machine runs for $n$ steps on an input of length $n$.
In fact, most halting Turing machines have unbounded running time: if the running time of a Turing machine is bounded by $T$, then the language it decides can only depend on the first $T$ symbols of the input.
If a Turing machine always halts on any input, then there is a function $f(n)$ such that the Turing machine always halts within $f(n)$ steps on an input of length $n$. You can take as $f(n)$ the maximal time it takes the Turing machine to halt on an input of length $n$. Since there are only finitely many words of length $n$ (recall that the input alphabet is fixed), this is well-defined.
Furthermore, if a Turing machine always halts on any input, the function $f(n)$ defined in the preceding paragraph is always computable. You can compute it by simulating the machine on all possible inputs of length $n$.
Space works exactly the same way.